Pro

NEW J E R S E Y

cedings of the 4th International ISAA Congress

York University, Toronto, Canada 11-16 August 2003

ings

Editors

H. G. W. BegehrtfREIE uNIVERSITAT bERLIN, gERMANY

r. p. gILBERTuNIVERSITY OF dELAWARE, uSa

m. e. muldoonoo nYorUniversity, Canada

M. W. WongYork University, Canada

London . Singapore . Beijing . Shanghai > Hong Kong . Taipi. Chennai

World Scientific

dvancesI n A n a l y s i s

Published by

World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Wmen Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

ADVANCES IN ANALYSIS Proceedings of the 4th ISAAC Congress

Copyright Q 2005 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereoj may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-256-398-9

Printed in Singapore by World Scientific Printers (S) Pte Ltd

Square Summability with Geometric Weight for Classical Orthogonal Expansions

Dmitrii Karp

Institute of Applied Mathematics Far Eastern Branch Russian Academy of Sciences 7 Radio Street Vladivostok, 690041 Russia dmkrp0yandex.ru

Summary. Let fk be the k-th Fourier coefficient of a function f in terms of the orthonormal Hermite, Laguerre or Jacobi polynomials. We give necessary and suf- ficient conditions on f for the inequality Ck 1 fk('ok < 00 to hold with 8 > 1. As a by-product new orthogonality relations for the Hermite and Laguerre polynomials are found. The basic machinery for the proofs is provided by the theory of repro- ducing kernel Hilbert spaces.

1 Introduction

The goal of this paper is to find necessary and sufficient conditions to be imposed on a function f for its Fourier coefficients in terms of classical or- thogonal polynomials to satisfy the inequality

m

k=O

with 0 > 1. So, we have three problems corresponding to the following three definitions of f k :

00

f k = /f(x)Hk(x)e-'ldz, (2) -03

and

408 Dmitrii Karp

-1

Here Wk, IL; and P ~ " is the k-th orthonormal polynomial of Hermite, Laguerre and Jacobi, respectively [19]. For the sake of convenience we use orthonormal instead of standardly normalized versions of the classical polynomials. In each case f is defined on the interval of orthogonality of the corresponding system of polynomials. We will use (Pk as a generic notation for either of the three types of polynomials.

Classes of functions with rapidly decreasing Fourier coefficients in classical orthogonal polynomials have been extensively studied. We only mention a few contributions without any attempt to make a survey. The series of papers [9]- [ll] by E. Hille is devoted to the Hermite expansions. Among other things, Hille studied expansions with fk vanishing as fast as exp(-~(2k + 1)'12). The functions possessing such coefficients are holomorphic in the strip IQzI < T.

Hille provided an exact description of the linear vector space with compact convergence topology formed by these functions where the set {Wk : k E WUO} is a basis. Its members are characterized by a suitable growth condition. In addition, he studied the convergence on and analytic continuation through the boundary of the strip.

is the interior of the parabola % ( - z ) ~ / ~ = r /2 . Rusev in [17] described the linear vector space with compact convergence topology formed by functions holomorphic in { z : % ( - z ) ~ / ~ < 7/2} where the set {ILL : k E N U 0) is a basis, and Boyadjiev in [4] studied the behavior of the Laguerre series on the boundary of the convergence domain.

If the coefficients (2) or (3) decrease as fast as exp(-.rks) with q > 1/2 the function f is entire. The spaces comprising such functions for Hermite expan- sions have been characterized by Janssen and van Eijndhoven in [12]. If the Fourier-Hermite coefficients fk decline faster than any geometric progression, the suitable characterization is provided by Berezanskij and Kondratiev in [2] (see Corollary 1.1 below). Byun was the first to study the Hermite expansions with condition (1) - see remark after Theorem 1. For the Laguerre expansions with lim supk IfkI'lk < 1 Zayed related the singularities of the Bore1 transform of f with those of F ( z ) = Ck f k z k in [20].

For the Legendre expansions with limsupk Ifk('lk = < < 1, Nehari relates the singularities of f on the boundary of the convergence domain { z : Iz + 11 + Iz - 11 < [ + 5-l) to those of F ( z ) = Ck f k z k in [16]. Gilbert in [7] and Gilbert and Howard in [8] generalized the results of Nehari to the Gegenbauer expansions and to expansions in eigenfunctions of a Sturm-Liouville operator.

Functions satisfying (1) apparently form a proper subclass of functions with limsupk lfkli 5 19-4. On the other hand the condition (1) itself cannot be expressed in terms of asymptotics of fk. Consequently, our criteria for the validity of (1) are of different character from those contained in the above

The convergence domain of the Laguerre series with fk N exp(-.rk 1/2)

Square Summability for Classical Expansions 409

references. Although they also describe the growth of f for the Hermite and Laguerre expansions and its boundary behaviour for the Jacobi expansions, our growth conditions are given in terms of existence of certain weighted area integrals o f f and cannot be expressed by an estimate of the modulus, while the restriction on the boundary behaviour is given on the whole boundary and not in terms of analysis of individual singularities.

2 Preliminaries

Throughout the paper the following standard notation will be used: N, R, R+ and CC will denote the positive integers, the real numbers, the positive real numbers and the finite complex plane, respectively. Since the coefficients (2)-(4) do not change if we modify f on a set of Lebesgue measure zero, all our statements about the properties off should be understood to hold almost everywhere. If, for instance, we say that f is the restriction of a holomorphic function to (some part of) the real axis, it means that f is allowed to differ from such restriction on a set of zero measure.

All proofs in the paper hinge on the theory of reproducing kernel Hilbert spaces (RKHS), so for convenience we briefly outline the basic facts of the theory we will make use of.

For a Hilbert space H comprising complex-valued functions on a set E, the reproducing kernel K(p , q) : E x E -+ C is a function that belongs to H as a function of p for every fixed q E E and possesses the reproducing property

for every f E H and for any q E E. If a Hilbert space admits the reproducing kernel then this kernel is unique and positive definite on E x E:

n

i,j=l

for an arbitrary finite complex sequence {ci} and any points pi E E. The theorem of Moore and Aronszajn [l] states that the converse is also true: every positive definite kernel K on E x E uniquely determines a Hilbert space H admitting K as its reproducing kernel. This fact justifies the notation H K for the Hilbert space H induced by the kernel K . The following propositions can be found in [l, 181.

Proposition 1 If H K is a Hilbert space of functions E -+ C and s is an arbitrary non-vanishing function on E , then

- K&, q) = s ( p ) s ( q ) K ( p , Q ) (6)

is the reproducing kernel of the Hilbert space HK, comprising all functions on E expressible in the form f s ( p ) = s(p) f ( p ) with f EHK and equipped with inner product

410 Dmitrii Karp

Proposition 2 Let El C E and K1 be the restriction of a positive definite kernel K t o El x E l . Then the R K H S H K ~ comprises all restrictions t o El of functions f r o m H K and has the n o r m given by

I I f l l l ~ K ~ = m i n { l l f l l ~ ~ ; f l ~ ~ = f i , f E H K ) . (8)

Proposition 3 If RKHS HK is separable and {$k : k E N} is a complete orthonormal system in H K , then i ts reproducing kernel is expressed by

where the series (9) converges absolutely for all p , q E E and uniformly o n every subset of E, where K(q,q) as bounded.

The relation K1 << K2 will mean that K2 - K1 is positive definite. This relation introduces a partial ordering into the set of positive definite kernels on E x E. Inclusion H K ~ c H K ~ and equality H K ~ = H K ~ will be understood in the set-theoretic sense, which implies, however, that the same relations hold in the topological sense as stated in the following two propositions. Proposition 4 Inclusion H K ~ c H K ~ takes place iff K1 << MK2 f o r a con- stant M > 0. I n this case M 1 / 2 ) ) f l ) l _> f o r all f E H K ~ . (Here llflll and 1 1 f 112 are the norms in H K ~ and H K ~ , respectively).

Proposition 5 Equality H K ~ = H K ~ takes place iff mK2 << K1 << MK2 f o r some positive constants m, M . In this case m1/211fl11 5 llfllz I M 1 ~ z ~ ~ f ~ ~ l .

Kernels satisfying Proposition 5 are said to be equivalent which is denoted by K1 M K2.

It is shown in [l] that the RKHS H K induced by the kernel K(p ,q ) = K1(p, q)K2(p, q ) consists of all restrictions to the diagonal of E' = E x E (i.e., the set of points of the form { p , p } ) of the elements of the tensor product HI = H K ~ @I H K ~ . The space HK is characterized by Proposition 6 Let K(p,q) = Kl(p,q)Kz(p,q) and let { $ k : k E N} be a complete orthonormal set in H K ~ . Then the R K H S H K comprises the func- tions of the f o r m

W 00

f(p) = c f L ( p ) $ k ( p ) , fl E HKi, llfk111? < 00. (10) k=l k=l

The n o r m in HK is given b y

where the minimum is taken over all representations o f f in the f o r m (10) and is attained o n one such representation.

Square Summability for Classical Expansions 41 1

3 Results for the Hermite and Laguerre Expansions

Functions satisfying (1) form a Hilbert space with inner product

00

k=O

This space will be denoted by Re for the Hermite expansions and by C i for the Laguerre expansions. The sets { 8 - k / 2 H k } k E N u o and {8-k/2L",k,Wuo constitute orthonormal bases of the spaces 'He and L i , respectively. For each space we can form the reproducing kernel according to (9):

The explicit formulae for these kernels are known to be [3]:

(13) (Mehler's formula) and

(Hardy-Hille's formula). Here I , is the modified Bessel function. The kernels (13) and (14) are entire functions of both z and 21. The space 'He comprises functions on R, while the space C i comprises functions on R'. Hence, we consider the restrictions of the kernels (13) and (14) to R and R+, respec- tively. Applying Proposition 2 with E = C and E' = R or E' = R+ we conclude that the spaces 'He and LE are formed by all restrictions to R and R+, respectively, of entire functions from the spaces generated by the kernels (13) and (14). We can drop minimum in (8) due to uniqueness of analytic extension and, consequently, the norm induced by inner product (11) equals the norm in HHK@ or in H , q .

Next, we observe that both kernels (13), (14) are of the form s ( z ) s ( ~ ) K ( z T i ) with non-vanishing functions S H ( Z ) = e-z2/(e2-1) and s ~ ( z ) = e-z / (e - l ) . Hence we are in the position to apply Proposition 1. In compliance with (7) the norms in HHK* and H ~ , Y are known once we have found the norms in the spaces induced by the kernels

-

and

412 Dmitrii Karp

(z..)-q/ (E) 2- - 1 / Ql//2+1 LK,(zii) = ~

0 - 1

Both of them depend on the product z;ii and are thus rotation invariant. Rotation invariance of the kernel implies radial symmetry of the measure with respect to which the integral representing the norm is taken. For the kernel (15) the measure and the space are well-known. It is the Fischer-Fock (or the Bargmann-Fock) space .To of entire functions with finite norms

where the integration is with respect to Lebesgue's area measure. By Propo- sition 1 the final result for the Hermite expansions now becomes straightfor- ward.

Theorem 1 Inequality (1) with fk defined b y (2) holds true for all restric- tions to R of the entire functions with

and only for them.

For inner product (11) this leads to the expression

(19) Since the polynomials Wk are orthogonal with respect to this inner product, we obtain the following orthogonality relation for the standardly normalized Hermite polynomials Hk:

Hk(z)H,(z)exp do = 6 k , , ~ ( 2 0 ) ~ k ! . 0 + 1 0 - 1

(20) These results for the Hermite expansions have been essentially proved by Du- Wong Byun in [5], although the emphasis in his work is different and it seems that the orthogonality relation (20) has not been noticed.

Corollary 1.1 Inequality (1) with fk defined by (2) holds true for all 0 > 1 iff f is the restriction to R of an entire function F satisfying IF(z)I 5 CeElzIz for all E > 0 and a constant C = C(E) independent of z .

The following corollary is an immediate consequence of (18).

Square Summability for Classical Expansions 413

A direct proof is given in [2]. For the Laguerre expansions the situation is a bit more complicated. The

kernel (16) is a particular case of a much more general hypergeometric kernel. The spaces generated by hypergeometric kernels are studied in depth in [15]. For the kernel (16) we get the space of entire functions with finite norms

where K, is the modified Bessel function of the second kind (or the Macdonald function). Application of Proposition 1 brings us to our final result for the Laguerre expansions.

Theorem 2 Inequality (I) with fk defined by (3) holds true for all restric- tions to R+ of the entire functions with

and only for them.

The orthogonality relation for the standardly normalized Laguerre polynomi- als L i that follows from this result is given by

Corollary 2.1 Inequality (1) with fk defined by (3) holds true for all 6 > 1 iff f is the restriction to R+ of an entire function F satisfying IF(z)I _< Ce“lzI for all E > 0 and a constant C = C(E) independent of z .

This corollary can be easily derived from (22) with the help of the asymptotic relation [3]

4 Results for the Jacobi Expansions

The space of complex-valued functions on ( -1 , l ) whose Fourier-Jacobi co- efficients (4) satisfy (1) will be denoted by J:@. Pursuing the same line of argument as in the previous section, we form the reproducing kernel of this space found by Bailey’s formula [3]

414 Dmitrii Karp

where 00

F4(a, b; c, c’; t , s) = c ( c ) n (c’ ) m rn !n !

m,n=O

is Appell’s hypergeometric function and ~ ( a : , p ) = 2a+P+1T(a: + l )r(p + l)/T(a: + p + 2). Define the ellipse E,g by

Ee = {Z : I Z - 11 + )Z + 11 < + F1I2}. (26)

Our first observation here is that both the series on the right hand side and on the left hand side of (24) converge absolutely and uniformly on compact subsets of Ee x Ee (see [13]). The implication of the uniform convergence is holomorphy of the kernel (24) in Ee x Ee with respect to both variables. Application of Proposition 2 with E = Ee, E’ = ( - 1 , l ) leads to the asser- tion that the space Jta,’ is formed by restrictions to the interval ( - 1 , l ) of functions holomorphic in Ee whereby the norm in &?’ equals the norm in H J K ~ , , due to uniqueness of analytic continuation.

Our main result for the Jacobi expansions will be derived from its particu- lar case a: = p = X - 1/2 whereby the orthonormal Jacobi polynomials reduce to the orthonormal Gegenbauer polynomials Ci. The reproducing kernel (24) reduces in this case to

2 ’ ( e 2 - z e Z E i q 2 ) 1 4e2(1- Z 2 ) ( i - ~ 2 ) eZx(e2 - 1) x + 1 A + 2 - - +q(e2 - 2ez;ii + i ) x + 1 2Fl ( T , T ; X + - *

(27) where 2F1 is the Gauss hypergeometric function and .(A) = & r ( X + 1/2)/r(X + 1). The series on both sides of (27) again converge absolutely and uniformly on compact subsets of Ee x Ee. Formula (27) can be obtained from (24) by using a reduction formula for F4 and applying a quadratic trans- formation to the resulting hypergeometric function. Details are in [13].

Let aEg denote the boundary of the ellipse Ee. We introduce the weighted Szego space AL2(aEe; p) with continuous positive weight p(z) defined on ~ E Q as the set of functions holomorphic in Eg , possessing non-tangential boundary values almost everywhere on dEe and having finite norms

Square Summability for Classical Expansions 415

We will write AL2(aEe) for AL2(aE8; 1).

thonormal Chebyshev polynomials of the first kind T k :

For X = 0, the orthonormal Gegenbauer polynomials reduce to the or-

c i ( z ) = T k ( Z ) = (2/T);Tk(z) = (2/7r)3 cos(karccOSz), I% E N, (29)

C:(Z) = T ~ ( z ) = ( l / ~ ) + T o ( z ) = ( l /n )+ ,

where T k is the Ic-th Chebyshev polynomial of the first kind in the standard normalization.

Lemma 1 The polynomials (Ok + 8-k))-1/2Tk/2 f o r m a n orthonormal basis of the space ALz(aE0; Jz2 - 11-i).

Proof The boundary aEe is an analytic arc which implies the completeness of the set of all polynomials in AL2(dEe; 1z2 - 11-a) (see, for instance, [S]). To prove orthogonality we will need some properties of the Zhukowskii function z = (w+w-l)/2. This function maps the annulus 1 < IwI < fi one-to-one and conformally onto E8 cut along the interval (-1,l) whereby the circle JwJ = fi corresponds to aE8. The inverse function is given by w = z + d n , where the principal value of the square root is to be chosen. We see by differentiation that the infinitesimal arc lengths are connected by the relation ldzl = 1w2 - l [ l d w [ / ( 2 [ ~ 1 ~ ) . Note also that z 2 - 1 = (w2 - 1)2/(4w2). Applying the identity Tk(z(W)) = wk + w - ~ we get by the substitution z = (w + w-’)/2:

e (m-k) /2 J-iu(k+m)dq = k # m , + 0-k, k = # 0 ,

0 k = m = O . +

27r

Combined with (29) this proves the 1emma.O Denote G A 0 - - Jx-1/2?A-1/2 0 . We are ready to formulate our main result for

the Gegenbauer expansions.

Theorem 3 Let X 2 0. The space 0,” is formed by all restrictions of the elements of AL2(aEe) to the interval (-1,l). The norms in Qt and AL2(dE0) are equivalent.

Proof The proof will be divided in three steps. Step 1. For the space HcK; induced by the kernel (27) with X = 0 we want to prove that

HGK: = AL2 (aE8). (30)

416 Dmitrii Karp

The weight 1z2 - 11-lI2 is positive and continuous on aEe so the norms in ALz(aE0; ( z2 - 11-4) and AL2(aEe) are equivalent and these spaces coincide elementwise. According to Lemma 1 and formula (9) the space AL2(8Ee; 1z2- It-$) admits the reproducing kernel given by

This kernel is equivalent to the kernel GK: due to (29) and inequalities

satisfied for any choice of n E N, ci E C and zi E Ee. Hence by Proposition 5 our claim is proved. Step 2. Consider the following auxiliary kernel:

7r-yo2 - 1) X + 1 X + 2 1 402(1 -z2) (1 - E 2 ) ;

+ 5; (02 - 20zE + 1)2 @ ( z , E ) = 2Fl 02 - 20zE + 1 (31)

It is positive definite as will be shown below. Substitution X = 0 yields'the identity

We want to prove that for all A, p > -f @(z,-iZ) = GK;(z,E). (32)

H q = Hk; . (33)

According to Proposition 5 we need to show that Kf x KE. Following the definition of the positive definite kernel (5), choose n E N, a finite complex sequence ci and points zi E Ee, i = 1,. Positive definiteness of the kernel [402(1 - z2)(1 - E 2 ) ] ' / ( 0 2 - 20zE+ 1)2k+1 due to its reproducing property in the Hilbert space of functions representable in the form f(z) = (1 - z2)'g(z), where g belongs to the Bergman-Selberg space generated by the kernel (36) with X = 2k + 1, and interchange of the order of summations justified by absolute convergence, lead to the estimates

Square Summability for Classical Expansions 417

where

- r(x + 1/2)T((X + 1)/2 + k)T((X + 2)/2 + k) > o. - r((x + 1)/2)T((X + 2)/2)F(X + 1/2 + k)k! This shows the positive definiteness of the kernel I?;. Using the asymptotic relation [3]

we obtain lim 4 = 1 + 0 < sup

k - m c uk kENo

The estimate from below is obtained in the same fashion with sup { u i / u ; }

replaced by inf {a i / a ; } . This proves equality (33). Combined with (32) and kENo

kENo (30) this gives:

H k ; = ALZ(aE8) (35)

for all X > -1/2. Step 3. According to (27) and (31), the kernel GK; is related to the kernel I?: by

GKf(2,E) = B ; ( z , E ) I ? f ( z , E ) ,

where

(36) For X > 0 the function B;(z ,E) is the reproducing kernel of the Bergman- Selberg space HB; [18]. This space comprises functions holomorphic in the disk ( z ( < [(02 + 1 ) / 2 ~ 9 ] ~ / ~ and having finite norms

where fk is the k-th Taylor coefficient of f. The functions

constitute a complete orthonormal system in Hq . Note further that the closed ellipse is contained in the disk IzI < [(02 + 1)/20]1/2 due to the inequality

418 Dmitrii Karp

J(O2 + l)/(20) > (04 + 0-+)/2, the right-hand side of which is equal to the large semi-axis of the ellipse Ee. As stated in Proposition 6 the space HGKi is obtained by restricting the elements of the tensor product HB; 8 HR; to the diagonal of Ee x Ee and comprises the functions of the form

By (35) we can put ALZ(dE0) instead of HA; here. For any g E AL:!(dEe) consider the estimate

IISTkIIAL2(aEe) 2 = /Ig(')Yk(')121d'I 5 zF$'e IYk(')1211SII~L2(aEe)' (37)

aEe

which shows that every product gyk belongs to ALz(aEe> and hence so does a finite sum of such products. Denote

M M

M M

Square Summability for Classical Expansions 419

Since both ck (/gk(IiLa(aEB) and c k [ a ! ( k ) ] 2 converge, the above estimates prove that the sequence Sn is Cauchy. It follows that HcK; c AL2(aEg). Inverse inclusion AL2(aEo)cHGK: is obvious, since I ( z ) = 1 belongs to HB; and so for any g ~ A L 2 ( a E o ) , the product Ig = gEHGKo. 0

Now it is not difficult to establish our main result for Jacobi expansions.

Theorem 4 Let a , P 2 -4. Inequality (1) with f k defined by (4) holds true for all restrictions to the interval ( - 1 , l ) of the elements of AL:!(aEo) and only for them.

Proof. Choose y > max(a,P}, then

JKz,P(z ,z) << MJK:,,(z,;iz)

for some constant M > 0. Indeed, for an arbitrary n E N, complex numbers ci and points zi E Ee, i = G, estimate using (24):

n

where

Interchange of the order of summations is justified by absolute convergence of the series (24).

Application of formula (34) yields as k, 1 -+ 00

aa'P A- 0 ( ( k + l ) a + P - 2 W - v - P ) = 0 ( ( 1 + Z / k ) " - - f ( 1 + k/Z)P--f) = O(1). -

k, l

Therefore sup {az,'//a:::} is positive and finite. Similarly by choosing -1 < q < min{a, P ) we can prove that

k,lENo

mJK;,, << JK&.

420 Dmitrii Karp

It remains to note that JK;,p = GKf, where X = ,L+ 1/2, and GKf is defined by (27). Now Theorem 3 gives the desired result. 0

When (Y and/or ,L belongs to (-1, -1/2), Step 3 of the proof of the Theo- rem 3 breaks down and the problem remains open.

Corollary 4.1 Condition (1) fo r the Fourier-Jacobi coefficients (4) of a funct ion f is satisfied for all 0 > 1 i f f f is the restriction of a n entire function to the interval (-1,l).

The last theorem and Szego’s theory [19] suggest that the following much more general conjecture might be true. Conjecture Inequality (1) holds true f o r the Fourier coefficients in poly- nomials orthonormal o n (-1,l) with respect t o a weight w that satisfies Szego’s condition f1 l n w ( z ) d z / d m > --oo i f and only i f f belongs to AL2(889).

Acknowledgments

The author thanks Professor Martin Muldoon and York University in Toronto for hospitality and support during the Fourth ISAAC Congress in August 2003 and Professor Saburou Saitoh of Gunma University in Kiryu, Japan, whose book [18] on reproducing kernels was the main inspiration for this research. This research has been supported by the Russian Science Support Foundation and the Far Eastern Branch of the Russian Academy of Science Grant 05- 111-r-01-046.

References

1. Aronszajn N. Theory of reproducing kernels, Trans. Amer. Math. SOC. 68, 1950,

2. Berezanskij Yu.M., Kondratiev Yu.G. Spectral methods in infinite-dimensional analysis. VoZ. 1, 2. Mathematical Physics and Applied Mathematics. 12. Dor- drecht: Kluwer Academic Publishers. xvii, 1995.

3. Bateman Manuscript Project (edited by A. ErdClyi), Higher transcendental functions, Vol. 1-3, McGraw-Hill Book Company, Inc., New York, 1953.

4. Boyadjiev L. I. On series representation in Laguerre polynomials. C. R. Acad. Bulg. Sci., 45, no.4, 1992, pp.13-15.

5. Byun D-W. Inversions of Hermite Semigroup. Proc. Amer. Math SOC. 118, no.

6. Gaier, D. Vorlesungen uber Approximation am Komplexen, Basel, Boston,

7. Gilbert R. P. Bergman’s integral operator method in generalized axially sym-

8. Gilbert R. P., Howard H. C. A generalization of a theorem of Nehari, Bull.

pp.337-404.

2, 1993, pp.437-445.

Stuttgart: Birkhauser Verlag., 1980.

metric potential theory, J. Math. Phys. 5 , 1964, pp.983-997.

Amer. Math. SOC. 72, 1966, pp.37-39.

Square Summability for Classical Expansions 421

9. Hille E. Contributions t o the theory of Hermit ian series, Duke Math. J. 5, 1939,

10. Hille E. Contributions t o the theory of Hermit ian series. II. T h e representation problem., Trans. Amer. Math. SOC. 47, 1940, pp.80-94.

11. Hille E. Contributions t o the theory of Hermitians series 111: M e a n values., Inter. Jour. of Math. and Math. Sci. 3, no. 3, 1980, pp.407-421.

12. Janssen A. J. E. M., van Eijndhoven S. J . L. Spaces of type W, growth of Henni te coefficients, Wigner distribution and Bargmann transform, J. Math. Anal. Appl., 152, 1990, pp.368-390.

13. Karp D.B. Square summability with geometric weight f o r Jacobi expansions, Far Eastern Journal of Mathematics, no.1, 2000, pp.16-27.

14. Karp D.B. Holomorphic spaces related to orthogonal polynomials and analytic continuation of functions, in: Analytic Extension Formulas and their Applica- tions, Kluwer Academic Publishers, 2001, pp.169-188.

15. Karp D.B. Hypergeometric reproducing kernels and analytic continuation f r o m a half-line., Journal of Integral Transforms and Special Functions, 14, 2003,

16. Nehari Z. O n the singularities of Legendre Expansions, J. Rational Mech. Anal.

17. Rusev P. Analytic funct ions and classical orthogonal polynomials, Sofia, 1984. 18. S. Saitoh, Integral transforms, reproducing kernels and their applications, Pit-

man Research Notes in Mathematics Series 369, Addison Wesley Longman, Harlow, 1997.

19. G. Szego, Orthogonal polynomials, AMS Colloquium Publications 23, 8th print- ing, 1991.

20. Zayed A. I. O n Laguerre series expansions of entire functions. Indian Jour. of Pure and Appl. Math. 12, no. 11, 1981, pp.1319-1325.

pp.875-936.

pp.485-498.

5, no. 6, 1956, pp.987-992.